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AFWL-TR.65-99 AFWL-TR 65-99
1/
BEHAVIOR OF
FLEXIBLE UNDERGROUND CYLINDERS
Ulrich Luscher
Massachusetts Institute off Technology
Department of Civil Engineering Cambridge, Massachusetts Contract AF 29(601)-6368
TECHNICAL REPORT NO. AFWL-TR.65-99
September 196 J CLEARINGHOUSE ^ FOR FEDERAL SCfEi'
Kardcopy j KioroZisb 7"^
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aj
AIR FORCE WEAPONS LABORATORY Research and Technology Division
Air Force Systems Command Kirtland Air Force Base
New Mexico
D D * • \ r"
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DiX^-iKA E
AFWL TR-65-99
BEHAVIOR OF FLEXIBLE UNDERGROUND CYLINDERS
Ulrich Luscher
Massachusetts Institute of Technology Department of Civil Engineering
Cambridge, Massachusetts Contract AF 29(60l)-6368
AFWL TR-65-99
POREWORD
This is one of two reports describing the research conducted at the Massachusetts Institute of Technology, Cambridge, Massachusetts, under Contract AF 29(60l)-6368 with the Air Force Weapons Laboratory (AFWL) between 1 March 196U and 2 April 1965. Lieutenant J. A. Eddings, AFWL (WLDC), was the project officer for the Air Force. The research was funded under DASA Project 5710, Subtask 13.157, Program Element 7.60.06.01.D. The report was submitted 10 August 1965. This work represents a logical continuation of the small-scale soil-structure interaction studies undertaken previously at M.I.T. and reported in the following reports and publications: Whitman, Luscner, and Philippe (l96l); Whitman and Luscher (1962); Luscher (1963); Luscher and Höeg (l96Ua); Luscher and Höeg (l961<b).
The research was performed in the Soil Research Laboratories of the Department of Civil Engineering at M.I.T. The Head of the Soils Laboratories and general supervisor of the project was Dr. T. W. Lambe, Professor of Civil Engineering. Dr. R. V. Whitman, Professor of Civil Engineering, contributed many valuable suggestions. Dr. U. Luscher, Assistant Professor of Civil Engineering, was Principal Investigator under this contract. This report was prepared by Dr. Luscher.
This technical report has been reviewed and is approved.
-tZtn^**' Or c!~d^L~~y*^
JAMES A. EDDINGS ILt USAF Project Officer
^uMiX 4^-7^ ROBERT E. CRAWFORI^/ ^/JOHN W. KODIS Major USAF " Colonel USAF Deputy Chief, Civil Chief, Development
Engineering Branch Division
ii
AFWL TR-65-99
ABSTRACT
An investigation was made of the "elastic" behavior and failure condition of underground flexible cylinders with particular attention given to arching, deformation and buckling. The report presents no nev data, rather draws heavily from experimental and theoretical work done in the past several years in an attempt to arrive at a unified picture of the chosen aspects of behav- ior. Active arching was found to reduce the load acting on tubes buried at depths up to several diameters in stiff soil by an average of 30 percent. On the other hand, passive arching may subject tubes buried in compressible soil to loads somewhat higher than applied on the surface. Spangler's defor- mation equation was modified to account for arching, lateral pressures, and variability of the soil modulus with pressure. Values of the modified modulus of passive soil resistance, back calculated by the new equation from tube deformation data, were successfully related to the constrained modulus of the soil. A comprehensive theory of buckling of underground cylinders is presented. It starts with the previously derived theory for elastic buckling in the circular-symmetric tube-soil configuration and extends it to cover (l) elastic buckling of an underground cylinder; (2) inelastic buckling; (3) the effects of soil stiffhess and presence of water; and (U) buckling of corrugated cylinders. It proved possible to correlate the soil modulus K8 controlling buckling to the constrained modulus M of the soil. The theory agreed well with the few available data. More comparisons with laboratory and field data are required, in particular to verify the values of K8 and their relationship to values of M. Regardless of the exact value of K8, however, it was shown that for many practiced situations of underground cylinders the controlling mode of failure is buckling rather than compressive yield.
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CONTENTS
Chapter 1
Chapter 2
2,1
2.2
2.3
2.1*
Chapter 3
3.1
3.2
3.3
3.^
3.5
3.6
3.7
Chapter U
k.l
h.2
REFERENCES
DISTRIBUTION
INTRODÜCTION
BASIC CONCEPTS OP BEHAVIOR OP FI£XIBI£ CYLINDERS
Introduction
Arching
Deformation of Flexible Cylinders
Failure of Flexible Cylinders
BUCKUNG OF FLEXIBLE ÜMDER5H0UND CYLINDERS
Introduction
Elastic Buckling of Soil-Surrounded, Staooth Cylinders
Inelastic Buckling of Soil-Surrounded, Smooth Cylinders
Buckling of Fully Burled Cylinders
Buckling of Shallow-Burled Cylinders
Bückling Resistance of Corrugated Cylinders
Effect of Soil Stiffness and Pore Pressure on Tube Buckling Resistance
SUMMARY AND CONCLUSIONS
Summary
Conclusions Regarding the Importance of Tube Bückling
Page
1
5
5
6
10
23
27
27
27
35
30
1*2
k6
U8
55
55
58
59
6?
UST OP JTGURES
Figure Page
1.1 Various Soil or Soil-Tube Configurations 2
2.1.a Correlation of Hoeg's Deformation Data Preliminary test series Ik
2.1.b Correlation of Hoeg's Deformation Data Main test series 15
2.2 Correlation of Donnellan's Deformation Data 16
2.3 Correlation of Marino's Deformation Data 17
2,k Correlation of Robinson's Deformation Data Id
2,5 Correlation Between E* and M for Dense Sand 2k
3.1 Tlieory for Buckling of Elastlcally Supported Ring 29
3.2 Coefficient of Elastic Soil Reaction for Elastic Ring 31
3.3 Buckling Strengths of Soil-Surrounded Tubes 32 Upper limit
3.4 Inelastic Buckling Curves 36
3.5 Correlation of M.I.T. Tube Buckling Data for Shallow Depth hk
3.6 Limiting Values of Constrained Modulus M 30
3.7 Effect of Pore Pressure on Buckling Resistance 53
vl
LIST OF TABIES
Table Page
2.1 Laboratory Data on Arching Around Buried Cylinders 8
2.2 Backcalculated E*-values from Laboratory Tests 13
3.1 Intersection of Elastic Buckling Curve with Yield Stress Line for Various Materials and Soil Stiffnesses 3**
3.2 Correlation of Donnellan's Destructive Tests '+5
3*3 Buckling Resistance of Corrugated Cylinders hj
vil
LIST OP SYMBOLS
A a coefficient
B dimensionless buckling coefficient
c stress-strain coefficient
D cylinder diameter
ü/t critical diameter-to-thickness ratio ' cr ft horizontal cylinder diameter n d depth of burial
E* modulus of passive soil resistance (units: psi)
E* modified modulus of passive soil resistance (milts: psi) E Young's modulus of soil s ^ E. tangent modulus of cylinder material
El flexural rigidity of cylinder wall
e modulus of passive resistance (units: psi/in.)
K coefficient of lateral pressure at rest
K coefficient of elastic soil reaction s M constrained modulus of soil
m value of M-functlon for p = 1 psi
N hoop resultant normal stress
n buckling mode
p vertical soil pressure acting on cylinder
p effective pressure at buckling in presence of pore pressure
p* elastic cylinder buckling pressure in circular-syranetric situation
p* critical buckling pressure at D/t cr cr p pressure applied at soil surface
p* surface pressure causing elastic buckling
p* surface pressure causing elastic buckling of deformed cylinder
R cylinder radlua
r. inside radius of elastic thick ring
r outside radius of elastic thick ring
vili
t thickness o*' cylinder wall
u pore pressure
ß arching factor
^ unit weight of soil
£A horizontal "strain" of cylinder
^av average stress acting on cylinder
0^ horizontal free-field stress In soil
(Tp peuaslve pressure on side of cylinder
CV vertical free-field stress In soil
6" elastic buckling stress ( = p* R/t)
G^ critical Inelastic buckling stress
G^ yield stress
V Polsson's ratio of soil
Ix
BLANK PAGE
.
—. -J IT-
CHAPTER 1
INTRODUCTION
The behavior of underground flexible cylinders le of Interest to
both the designer of underground pipelines or culverts and the designer of
prott»ctlve structures. A considerable amount of research on the topic has
been done by both groups of users. While originally the Interests of the
tvo groups went in different directions, a rapprochement has recently taken
place, in the sense that tne two groups have learned to interact with and
profit from each other. An outward example of this new cooperation was the
Symposium on Soil-Structure Interaction held at the University of Arizona
In June 19^^, where representatives from both groups came together to dis -
cuss common problems.
There is much evidence of this new-found cooperation in the recent
literature. For instance, workers In protective construction have begun
to appreciate more and more the classical work done on burled pipes and
arching by Marston, Spangler and Terzaghl. On the other hand, workers
with conventional pipe installations, faced with ever-increasing loads
from high embankments for highways and dams or from heavy live loads such
as airplanes, have come to realize that they derive benefits from the work
on high-resistance Installations done by workers concerned with protective
construction.
The work done at the Massachusetts Institute of Technology over
the past six years has been an attempt to introduce conventional soil
mechanics knowledge and procedures into the study of the behavior of
structures used In protective construction. This report represents a
continuation, and in many ways a conclusion, of the work on soil-surrounded
flexible cylindrical structures. The philosophy of the approach taken was
to start with simple situations with respect to geometry and load applica-
tion and obtain a thorough understanding of these before progressing to
more complicated situations. Consequently the work progressed from hollow,
thick-walled cylinders of soil (Fig. 1.1a) to circular-symmetric soil-tube
:
T - -
rubber membranes flexible tube
Pi (on tube)
(a) Hollow Soil Cylinder (b) Soil-Surrounded Tube
membrane
soil
flexible tube
(c) Buried Tube
FIG. 1.1 VARIOUS SOIL OR SOIL-TUBE CONFIGURATIONS
—
configurations (Fig. 1.1b) to tubes buried horizontally below a soil
surface (Fig. 1.1c). Consideration of dynamic loading instead of the
static loading used heretofore would be r last step. This step is, how-
ever, not being planned at present, chiefly because the large-scale effort
required seems more suitable for a research organization than for a univer-
sity.
The last report on the progress of the research (Luscher and
Höeg, 196^) was concerned primarily with the "elastic" behavior and the
collapse condition of a flexible structural tube symmetrically surrounded
by soil (Fig. 1.1b). Some preliminary results from tests on flexible
buried cylinders were also presented.
The present report is concerned with the behavior of flexible
tubes mainly in the "buried" condition and presents an overall picture
of certain aspects of cylinder behavior - arching, deformation, and buckling.
The material is based on the experimental and theoretical work presented
in the 196^ report mentioned above, plus pertinent information from the
literature. This report thus contains no new experimental data, but rather
attempts to arrive at a unified picture of the chosen aspects of behavior
on the basis of a synthesizing study of existing data and theories,
A companion report prepared by K. Höeg (19^5) describes a new
study of the interaction between underground structural cylinders and
the surrounding soil. A mathematical formulation considers soil to behave
as a continuous, elastic material. The applicability of the analytical
approach was tested in the experimental phase of the study, with experi-
ments designed to measure directly the contact pressures between sand and
buried cylinders. The variables were cylinder flexibility, cylinder com-
pressibility, depth of sand cover and level of applied surface pressure.
The condition specifically investigated in the present report is
that of a long, flexible, cylindrical tube buried in horizontal position
below a horizontal soil surface and loaded by static pressures applied on
that surface (Fig. 1.1c). If the depth of burial is at least "full,"
corresponding to a depth of cover of one to two tube diameters, this case
Is no different from the case of a tube loaded by a high earth fill.
Chapter 2 is concerned with a number of aspects of the behavior
of buried flexible cylinders - arching, deformation, failure condition.
The infomation was gathered as much for its own sake as to provide Inputs
for the subsequent chapter. Chapter 3> which treats buckling of burled
flexible cylinders. This work on buckling represents an extension and
conclusion of the earlier research by the author (Luscher and Höeg, 196^).
It extends the theory for buckling of soil-supported tubes from the clrcular-
synnnetric situation to the buried-tube situation, into the inelastic range,
to buckling of corrugated pipes, and to more general conditions of soil
surrounding. General conclusions are reached about the Importance of buckling
as a possible failure mode for flexible buried cylinders.
&•
CHAPTER 2
BASIC CONCEPTS OP BEHAVIOR OF FI£XIBLE CYLINDERS
2.1 INTRODUCTION
This chapter quantitatively evaluates certain aspects of the
response of burled cylinders. The Information was gathered from numerous
recent publications concerned with cylinder behavior. The purpose Is to
set the stage for the subsequent chapter on buckling, by relating buckling
failure to the other possible modes of failure and by providing needed
numerical Inputs on arching and tube deformation. Thus this chapter Is
not meant to cover all aspects of the behavior of flexible cylinders, but
rather to present information of general interest collected in connection
with the investigation of buckling.
The free-field stresses in a dry soil mass are a vertical stress
(T = yh + p and a horizontal stress ^. = K (T , where Vh is the v o h o v7 " weight of the soil surcharge, p is the uniform pressure applied at the
soil surface (over a large area), and K Is the coefficient of lateral
pressure at rest. In general K might vary from 0.35 to 0.7 for most
soils except heavily overconsolldated clays. If an inclusion, in the
present case a burled cylinder, deformed Just like the soil, it would be
exposed to these stresses. However, a flexible cylinder has only a limited
capacity to withstand these highly uneven stresses and will start to deform
into a shape resembling a horizontal ellipse. In so doing it mobilizes
lateral passive soil stresses which counteract the deformation. Under in-
creasing applied loads the cylinder will further deform in such a way as
to carry the load most efficiently; it will be exposed to large hoop com-
pression stresses, but the bending mooents will be small compared to the
bending monients which would be required to carry the free-field stress
system with no soil support. Eventually the cylinder will fall in one of
several possible failure modes.
The important questions associated with loading of the cylinder-soil
! '
system are:
1. How is the hoop stress related to the free-fie Id
stresses, i.e., what is the arching condition?
2. What is the deformation of the pipe?
3. What is the condition of failure?
The following sections will discuss these three questions in turn.
2.2 ARCHING
How much stress, in relation to the free-field stress system,
reaches the buried cylinder? It is clear in this context that not the
extremely low flexural rigidity of an unsupported cylinder, but rather
the much higher vertical rigidity of the soil-supported cyllner, or pos-
sibly even the corapressive (volumetric) rigidity, should be compared to
the rigidity of the soil when investigating arching.
Arching was studied by Luscher and Hoeg (196^) in connection with
tests on clrcular-synmetric tube-soil configurations (Fig. 1.1b). It was
found for the particular situation investigated that the stress carried
by thin-walled aluminum tubes was within ± 20 percent of the stress applied
on the outside of the soil cylinder, with individual values depending on
soil density and soil-ring thickness. "Hius, only little arching action
tocjc place. The experimental result agreed with the developed elastic
theory, which indicated that the compressibility of the tube in the pure
compression mode (mode zero) was similar to the compressibllty of the sur-
rounding sand ring. In tests using plastic tubes with much higher compres-
sibllty, on the other hand, very effective active arching w^s mobilized
in the sand ring; the share of the applied stress carried by the tube
dropped to as little as 20 percent.
For tubes in a burled configuration (Fig. 1.1c), one might expect
the horizontal ellipslng, which is associated with vertical shortening,
to lead to a reduction in vertical applied stress on account of arching.
.
If without deformation the vertical stress is T and the horizontal v stress K CT . then the more-or-less unifonh stress G~ after defonna- o v av tion might be expected to be the avervge of the tvo, or ^(l + K ) J" . This
type of arching, which vas called "pressure redistribution" by Luscher and
Höeg (1964b), thus derives from the increase in vertical compressibility
due to ellipsing of the tube. Since the ellipsing is the principal effect
which distinguishes the buried-tube case from the circular-symmetric case,
no further appreciable arching effects (beyond pressure redistribution)
would be expected for flexibli metallic tubes, where the compressibility of
the tube in the zero mode is roughly comparable to the compressiblity of
the surrounding soil.
Evidence exists that the above concepts are correct, at least for
dense sand. Preliminary test data reported by Luscher and Höeg {196ha)
indicated that the applied surface pressure at buckling of a tube in a
rigid box loaded only at the surface (K stress system) was 50 percent
higher than the surface pressure of a tube in a box loaded equally in
vertical and horizontal direction, (if K = 0.4, i(l + K ) = O.J. Thus a
tube in the first situation "feels" only JO percent of the surface pressure,
while a tube in the second situation "feels" 100 percent. If the stress
leading to tube failure is the same in both cases the surface pressures at
failure can therefore be expected to be in the proportion 10 to 7, which is
close to the observed If to 1.).
A considerable body of evidence from tests on flexible, metallic
tubes of h- to 6-in. diameter, buried at a depth of one-half to two tube
diameters in dense sand is suramariaed in Table 2.1 (Luscher, 19^5; Donnellan,
1964; Marino, 1963; Robinson, 1962). The tubes were strain-gaged inside
and out, making it possible to determine hoop stress resultants. The
value of the measured hoop force on the sides, indicative of the total
vertical load carried by the tube, varied between 33 and 100 percent of
applied load, with an average just under 70 percent. The average of the
hoop forces at top and bottom, indicative of the total horizontal load
carried by the tube, was somewhat lower than the force on the sides and
averaged 50 percent of applied loads. The depth of cover within the quoted
limits had very little effect upon these values, as shown clearly by
Donnellan's (1964) data. The variation in D/t over the range 80 to 120
8
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and the tube material also had only small effects (Robinson, 1962). Bie
major variable seems to have been a personal one, evidenced probably
mainly in the soil density, but also in the method of placement of the
tube, test device used, instrumentation etc. The only known data of this
kind for loose sand (Luscher, 1965), also presented In Table 2.1, indicated
side hoop forces up to 50 percent higher than the applied load, and averaged
top and bottom hoop forces ho to 60 percent of applied load. Hoeg (1965)
presents data from tests with direct measurement of contact stresses which
show that the average stress applied on a steel tube with D/t = 80 buried
at ID depth in dense Ottawa sand was about 70^ of the applied surface
pressure.
It can be concluded that for flexible, metallic tubes buried in
dense sand, the hoop forces are somewhat smaller than the vertical applied
load on account of press are redistribution and arching. This reduction
has been observed to be anywhere between 0 and TO percent, with an average
of about 30^. For loose sand, the little available evidence indicates
no such beneficial action, rather the possibility of some increase in
pressure acting on the tube due to passive arching. These statements can
be summarized in the highly approximate equation
P = Po//3 (2.1)
■where /3 = 1.5 for stiff soil and 1.0 for compressible soil.
Hie above concepts of loads acting on underground cylinders are
extremely crude. Their only merits are that they are simple, and that
they originate from - however limited - experimental evidence. Marston's
and Spangler's work (Spangler, i960) resulted in a complete system of
recommendations for loads on buried conduits. However, these theories
were derived and verified primarily for rigid pipes and are therefore
unacceptable here in connection with flexible cylinders. More recently,
several arching theories have been developed based mainly on the concept
of vertical slip surfaces (e.g., Newmark and Haltiwanger, 19^2; Finn,
1963; Allgood, 1965). Use of any of these theories would require additional
numerical Inputs, the end result could not be expressed In the simple
form of Eq. (2.1), and It Is not even certain that the result would be
any more trustworthy.
2.3 DEFORMATION OF FLEXIBLE CYLINDERS
2.3.1 Deformation Equation
The standard method to predict deformations of burled cylinders
Is by way of Spangler's (i960) equation, which may be written as
eu = —2— = 5 , (2.2) h D 85 El/D3 + 0.65 E'
where 6^ = horizontal "strain" h
AD. = change In horizontal diameter
D = cylinder diameter
El = flexural rigidity of cylinder wall
p = vertical soil pressure acting on cylinder
E' = modulus of passive resistance of soil.
The coefficients In the denominator were calculated for a deflection lag
factor of 1.0 and a bedding constant of 0.09^, corresponding to a bedding
angle of 50°.
This equation appears to be basically sound. However, there are
a number of difficulties associated with Its use. One of the difficul-
ties Is the choice of a value for the applied pressure p, which is usually
calculated by the Marston-Spangler theories of loads on underground con-
duits (Spangler, i960) and involves a number of critical assumptions.
Further, in the derivation of the equation the fact was neglected that
even without any deformation the lateral pressure would not be zero, but
K (T • Thus, considering also pressure redistribution as discussed in the w V
preceding section, the vertical pressure which has to be resisted by the
combination of the tube rigidity and lateral passive pressure
10
IF not (J , but rather (for dense soil)
^•v ~Ko%- Ü1 + Ko) ^v ' Ko [\ = i^1 - Kc) ^v-
The primary difficulty with the use of Eq. (2.2) is associated
with the choice of E'. Reconmendations for this choice have been formu-
lated (ASCE, 196^). However, such "handbook" values can only be crude
estimates since crude soil identification is used rather than tests on
the actual soil, and consequently a high factor of safety is required.
Furthermore, these reconmendations are for constant E'-values, and thus
cannot predict the curved loeui-deformation relationship frequently observed.
To Improve on this situation, an attempt was made to correlate tube
deformations measured in several recent tests and test series with the
theory, i.e., to backcalculate E1-values from experimental data. First
Eq. (2.2) had to be modified according to the ideas presented above. The
following assumptions were made for this modification:
1. To account for pressure redistribution and arching as well as
at-rest lateral pressure, p in the equation is set equal
to 1/3 ^ (which is about equal to 5(1 - K ) fr.) for dense
soil, and i (7 for loose soil. ' ^ v
2. If E1 is variable with pressure, the equation is applied
incrementally, relating A£ to rf3 i the total horizontal
strain £. is then the sum of all increments up to the
pressure under consideration.
With these assumptions, the deformation equation becomes
- 0.5/0- AQ"V A6h = X 1 (2.3) 85 El/ir + 0.65E*
where /3 = 1.5 for dense soil (0.5/p = l/3)
1.0 for loose soil (0.5/(3 = l/2)
E* = f( CT ) is a modified modulus of passive resistance of soil
(defined differently because of the modification in the load term),
11
—
The form of the E*-functlon will In general be assumed as a power function,
E* = C ()vb. If the tube rigidity tenn 85 Ei/D3 IS negligible compared to
the soil term 0.65E#, the expression for ä£ can be Integrated to give a
direct solution for L as a function of (T • On the other hand. If the
tube rigidity Is considered and E* Is a function of G", a numerical, step-
by-step solution Is best used.
2.3,2 Experimental Evidence
Recent tube deformation data from various sources were used to
determine, by substitution Into Eq. (2.3), the function E* = C CT which
best satisfied the experimental load-deformation curves. In some cases
where only final deformations but no load-deformation curves were avail-
able, only constant values of E* could be determined. In two cases where
the load-deformation curve was a straight line over the region of Interest,
a constant value of E* was also calculated. In all other cases locking
behavior was apparent. I.e., the rate of tube deformation decreased with
Increasing applied pressure. While every curve could be fitted with one
best E*-functlon, with em exponent b which might lie anywhere between zero
and one depending on the character of the curve, It became soon apparent
that a square-root function
E* = C G" ^
gave the best fit overall. Thus each of the available loeui-deformation
curves was fitted with a theoretical curve based on an E*-functlon of this
type.
The results of fitting several quite similar laboratory experiments,
all Involving tubes of h- to 6-lnch diameter burled at a depth of at least
D in dense sand, are presented in Able 2,2 and Figs. 2.1 through 2.k (Hoeg,
I965; Donnellan, I96U; Marino, 1963; Robinson, 1962). The table and figures
show that all four series of tests gave amazingly consistent E*-values of i 1 1
between 1900 p? and 3000 p2, with an average of about 2^400 p2. The one- o o o
half power law gave on the average the best fit, as it resulted in load-
deforaatlon curves which were "best-fit" in seme cases, too much curved In
12
£T
.'■III ■IIIIH^
0.006
0.004
0.002
150
P " PSl
FIG. 2.1. a CORRELATION OF HOEG'S DEFORMATION DATA
Preliminary Test Series
1^
0.006
0.004
0.002
' o
FIG. 2. l.b CORRELATION OF HOEG'S DEFORMATION DATA
Main Test Series
15
~ .... - *■ *
0.004 1 D/t = 114, Aluminum 6061 - T4
Depth 1.5- 2D (average)
0. 002
0
Calculated with * l/2 E = 3000p ^
Observed
0 50 100 150
P " PSi o
FIG. 2.2 CORRELATION OF DONNELLAN'S DEFORMATION DATA
16
0. 01 D/t = Hü, Steel
Depth 2D
Ü.0Ü5
Calculated with * l/2 E =2200p '
0 100 200 300 400
Po - psi
FIG. 2.3 CORRELATION OF MARINO'S DEFORMATION DATA
IT
0.004
0.002
D/t = 80, 120
Steel
Calculated with * l/2 E = 2500 - '
D/t = 120 (observed)
D/t = 80 (observed)
100
Po - ps.
0.004
D/t ^ 120
Aluminum
0.002
(Sat 15c from horizontal)
v Calculated with
* 1/2 E = 2700 p o
50 100
P - PS» 'o
FIG. 2.4 CORRELATION OF ROBINSON'S DEFORMATION DATA
10
others and not enough curved In still others. Thus, one can sunnnarlze
by saying that a valur of at least
i E* = 2000 § 2 (2.10
vas reached quite consistently in lahoratory tests using carefully placed,
dense sand.
Allgood (1965) presents data on a steel tube of 2U-in. diameter
and D/t = 500 buried at 3/8 D depth in "dense" Monterey sand. The straight
load-deformation curve in the pressure range 12-25 psi (well beyond
a previously applied pressure of 10 psi) resulted in an E*-value of 38OO
psi. Over the same range of pressures the experiments sunmarized in
Table 2.2 gave B»-values between 7500 and 12,000 psi. Allgood's smaller
value must be due to the smaller depth of cover and probably a smaller
relative density of the sand; the latter possibllty is quite high in view
of the large soil volume that has to be placed in the test pit of the NCEL
atomic bla'it simulator.
An example of the effect of backfill density in laboratory studies
is given by Luscher's (19^5) data from dynamic tests on buried aluminum
tubes of 4-ln. diameter. The results were:
dense sand (relative density 90^): E* ~ 2100 (T*
loose sand (relative density 30^): E* = 950 (L •
Effects other than soil density may have had an Influence on the relative
values of the deformations in dense and loose sand (e.g. arching, or
different dynamic effects). However, direct comparisons of this kind are
rare enough that it was felt these data should be Included here. Further-
more, the excellent agreement of the E*-values for dense sand with the
data in Table 2.2 indicates that for this situation the dynamic character
of the loading did not alter the deformations appreciably. It can reason-
ably be assumed that the data for loose sand are Just as representative
for that density.
Bulson (1962) presents deformation data for very flexible steel
tubes with D = 10 in. and D/t = 667, buried at a depth of 3A D in
19
"compacted sand" vhich was probably of medium to low density. The result-
ing E* was almost constant at HOC psl with pressure up to Uo psl. There
are, however, uncertainties associated with rigid-body motion, with the
ratio of horizontal to vertical deformation, and with the relative density
of the sand which make this value highly tentative.
The remaining correlations are for field case studies. Since
In all these cases no load-deformation curves but only final deformations
were available, constant values of E* were backcalculated.
E*-values for field Installations of pjpes of 30-ln. diameter and
0.312-ln. wall thickness can be obtained from Lambe (i960, 1963). For
two different pipes Installed without any control of the granular backfill,
the values were 1100 and 750 psl. For pipes carefully backfilled to mini-
mize deformations and loaded by a fill of 75-ft. height, on the other hand,
E* was backcalculated to be at least 7000 psl.
Barnard (1957) quotes some data on two Instrumented pipe situa-
tions. The first of these, a smooth pipe with 30-ln. diameter and 0.109-ln.
wall thickness, showed a horizontal diameter change of 0.32 in. under 12
ft. of "tamped-sand" fill. The resulting E* for this probably quite loose
fill was 600 psl. The second case, a 1-gage multi-plate pipe of 7-ft.
diameter under 137 ft. of fill, yielded E* = 2700 psl for a carefully
placed backfill. These two E*-values are probably quite typical of the
possible range in field installations.
When comparing these backcalculated values of E* with published
values of E', it should be borne in mind that the E*-values were obtained
from Eq. (2.3) and therefore are reduced by a factor of 2 to 3 compared to
E' on account of the initial assumptions made (arching, lateral at-rest
pressure). Thus E*-values should be multiplied by a factor 2 to 3 when they
are compared to published or generally accepted values of E'; inversely,
published values of E' should be reduced by that factor when they are to
be used as E* in Eq. (2.3).
These coraaents apply to Watklns's (1959) work, which led to E'-values
(calculated by Eq. (2.2)) of 1500 to hOOO psl for laboratory tests on mixtures
20
of silt and clay of varlble known density, but unknown water contents.
On the other hand, values obtained In the so-called Modpares device
are direct measurements of the modified passive soil modulus E* and should
therefore be directly comparable to the values backcalculated here. ASCE
(196^) contains such values, which are "suggested as a guide" (quoting
from publication) and contain a built-in safety factor of as much as 2.
The values vary between 720 and MOO psl for sand at various densities,
and between 51+0 and U020 for clay at various densities.
The Information on values of E* backcalculated from test results
and quoted In the literature may be summarized as follows:
1. For granular soil, carefully placed under laboratory con-
ditions to achieve maximum density, the load-deformation
curve exhibits locking character. This is expressed by
an E* which increases with a positive power of the applied
pressure. The conservative relationship
i E* = 2000 (J 2 (E* and () in psl) (2.4)
describes well the results of several recent test programs.
It corresponds to average values of about 3000, 7000 and lU,000
psl for the pressure ranges 0-10, 0-50 and 0-200 psl, respectively.
2. Very little evidence exists for laboratory E*-values of
loose granular soil and cohesive soil of any consistency.
Values ranging from 700 to 3000 psl for loose sand, and
from 500 to 6000 psl for cohesive soil are mentioned in the
literature.
3. In field Installations with well-controlled backfill, E*-
values of 3000 psl or higher have been achieved.
k. For field Installations with very little or no backfill
control, values of E* in granular soil appear to be at
least 500 psl. No corresponding minimum value could be
established for cohesive soil.
21
It should be pointed out again that all values quoted above are
for use with Eq. (2.3), For use with Eq. (2.2) the E»-values should be
Increased by a factor 2 to 3. Eq. (2.3) Is believed to be preferable
to Eq. (2.2) since It considers arching and lateral at-rest pressures.
2.3.3 Correlation of E* with Compression Test
The remaining question Is: Can the E*-values quoted In the
preceding section be correlated -uo any soil property obtainable In a
standard soil test? The locking behavior observed in many cases Indi-
cates that the correlation should be made to the one-dimensional com-
pression test rather than the trlaxlal test. Therefore, the laboratory
data for granular soils will be compared with data from one-dimensional
compression tests on Ottawa sand, taken from Fig. U,3 of Luscher and Hoeg
(1964a) and reproduced as Pig. 2.5 of this report.
To undertake this correlation between E* and the constrained modulus M, a theoretical relationship must be established between the
two moduli. E* can be expressed as
E* = eR,
where e Is the modulus of passive resistance, In units of psl/ln. Now
e can be related to M by a simple pressure-bulb consideration. The lateral
passive pressure Is usually assumed to act with parabolic distribution
over the middle 100° on the sides of the pipe (Spangler, i960). As equiva-
lent "footing width," two thirds of the chord subtended by a central o / angle of 100 is used, or 0.51 D ( - width of uniformly loaded rectangular
strip with peak pressure equal to that of a parabolically loaded strip). A reasonable depth of an equivalent pressure bulb with triangular stress
distribution has been found to be 3^25 tinea the width of the loaded area
(Barnard, 1957), or 1.6? D. Then,using E. = 0.75 M (Luscher and Hoeg, 196^) s as modulus in the presrure bulb.
Aft 0 1.6? D
2 2 0.75 M
22
>
Thus *V2
1.5 M
1.67 D
which leads to E* m 0.45 M.1'
Accordingly, the laboratory E*-value8 for granular soil from the
preceding section were multiplied by a factor 2.2 to obtain values of M,
vhich were then plotted In Fig. 2.5. flie curves were not plotted In the
lowest pressure range because the observed load-deformation curves and
consequently the backcalculated E*-values axe poorly defined in that
region. It was found that the relationships for both dense and loose
sand agreed reasonably well with the corresponding values obtained in one-
dimensional compression tests.
Thus the conclusion is reached that the modified modulus of
lateral pressure E* can, at least for granular soil, be correlated with
the one-dimensional modulus M:
B* » 0.^5 M. (2.5)
This relationship was derived theoretically by a simple pressure-bulb
consideration and agreed well with experimental data.
2.U FAILURE OF FLEDOBIE CYLTNDERS
The possible structural failure modes of a flexible cylinder burled
1) Note that this result is somewhat different from Barnard's (1957)» which would be E* = 0.3 M. Barnard used the full width of the chord as "footing width", while here a factor 2/3 is added to account for the parabolic stress distribution. The same result of E* « O.U5 M would be obtained if Barnard's "footing width" were used and the applied uniform pressure reduced by a factor 2/3. Exact numbers are not Important here in view of the generally approximate nature of the calculations; the Im- portant point is that these considerations indicate that E* should be roughly 2 to 3 times smaller than M.
23
r
ns
Constrained Modulus M - 10 psi
T
p
n c S3 w r >
tr M H ? M M Z
M
> 2 0
D m "Z w M W > 2:
n c
3
5 H o
c -5 ft
x
at a depth of at least one diameter are:
1. Joint failure
2. Excessive deformation leading to cavlng-ln of the crown
of the tube.
3. Elastic buckling of the tube wall under hoop stresses
vhlch are excessive for the tube rigidity and lateral
support provided.
h. Yielding of the tube wall due to excessive hoop stresses,
resulting In general crushing unless such a failure Is closely
preceded by plastic buckling due to decrease In wall rigidity.
Of there failure modes the last one Is most desirable, since It
represents most efficient use of the structural materials and will, for
a given situation, take place at the highest pressure. The other modes
are forms of premature failure that should in general be avoided, If
possible.
Failure by excessive deformation (be It defined as above, as
collapse Induced by excessive deformation, or aa failure to function
satisfactorily due to excessive deformation) can generally be avoided
by appropriate control of the backflU on the sides of the cylinder.
Cases vhere this presents difficulties, and where deformation may thus
remain the controlling factor, are l) bored tunnels, where no control
over the backfill is possible, and 2) cases where the allowable deforma-
tions are small because of the function of the cylinder (e.g., carrier
pipes, as discussed in Luahe, i960). The situation with respect to
deformations can frequently be improved by installing the cylinder with
vertical "elllpsing," i.e., vlth the vertical diameter somewhat Increased
by fabrication or by installation of struts.
Premature Joint failure can always be avoided by constructing
joints to at least the full strength of the cylinder wall.
Buckling is the remaining possible mode of premature failure.
25
It Is well known that a flexible cylinder loaded by outside liquid
pressure has low buckling resistance. When the stress on the cylinder
Is applied through soil, however, the danger of buckling Is greatly
reduced on account of the shear strength of the soil. It Is of great
practiced Interest to know by how much the buckling resistance Is In-
creased. This topic Is discussed In the following chapter.
26
CHAPTER 3
BUCKUNG OF FLEXIBLE UNDERGROUND CYLINDERS
3.1 INTRODUCTION
In this chapter a solution will first be presented for the most
basic case of buckling of a soil-surrounded tube: that of elastic buckling
of a circular tube with a surrounding thick cylinder of one specific soil
and loaded by uniform radial pressures. The theory for this case has
been developed quite recently and has been verified experimentally for
a limited range of tube stiffnesses and soil conditions.
In subsequent sections deviations from this most basic case are
investigated. These extensions of the buckling theory were derived
theoretically, and were verified by existing experimental evidence where-
ever possible. The specific cases considered were the following: Section
3.3 discusses inelastic buckling. Sections 3»^ and 3'5 treat the effects
of burial, i.e., of the change in geometry from circular-syranetric to
underground with load applied at the surface; Section 3•,* is concerned
with the "fully-buried" case where the effects of the soil boundary are
negligible, while Section 3.5 investigates the effects of a near-by soil
boundary (shallow burial). The buckling resistance of corrugated pipes
is studied in Section 3«6. Finally, Section 3.7 explores the effects of
soil stiffness and of the presence of pore pressure on the buckling re-
sistance.
3.2 ELASTIC BUCKLING OF SOIL-SURROUNDED. SMOOTH CYLINDERS
The theory for elastic buckling of an elastlcally supported
cylinder predicts instability under an average applied outside pressure
p* of
p« = 2 V S, , (3.1)
2?
vhere K = coefficient of elastic soil reaction. 8
For a derivation of this equation, see Link (1963), Luscher and Hoeg (196^a),
or Allgood (1965). ' The relationship is plotted in Pig. 3.1 for a fixed
value of the term Ei/R = 1.0, corresponding to a flexible tube (D/t = 280
for steel). The theory further indicates (Luscher and Hoeg, 196^a) that
the critical buckling mode n is
n = 1(—S . (3.2)
2) The buckling modes are also indicated in Pig. 3.1 '
The task of predicting the critical buckling condition is thus
reduced to finding the coefficient of elastic soil reaction K . Por the
simple geometry of a flexible tube surrounded synanetrically by a thick
cylinder of sand and loaied by uniform, radially acting pressures (Pig.
1.1b), Luscher and Hoeg {I96ka) showed that K backcalculated from experi- s mental failure data was equivalent to the coefficient of resistance of
the soil ring to uniform, outward acting pressure in the cavity. Thus K 8
was dependent upon the ring geometry and the "elastic" soil properties.
Since these properties are not constants in a given soil but depend on
the stress level and stress distribution, K is affected not only by the 8
pressure p-* itself but also by the arching condition in the soil ring.
Only for those ccoblnations of soil and tube for which the arching effect
is small can the modulus K be expressed as a function of p* alone, then s the buckling equation solved for p*.
An equation of this kind was obtained by formulating
K8 = B E8, (3.3)
where E = Young's modulus of the soil s
' Equations (3«l) *nd (3.2) were derived under the assumption that n is em algebraic rather than an integer number. Consequently they do not give correct results for extremely low values of K where n is small. However, it can be shown that the error of Eq. (3.1) is practically negli- gible for n ? 3.
28
a
tn
c 0
Ü ed
o
ea w
c
V c U
100,000
;?■■ 1 0
1
n = 17. r 50,000
"
20,000
n r 10 10.000 - 1
- 1 .1000
■■
/
2000
n = 5 .6 / 1000
-
/
/
500
•-
/ 200
/
100 n = 3. 1 / « • . 1111 . __ L J. i 1 i
10 20 50 100 200 500 1000
Elastic Buckling Pressure p - psi
FIG. 3.1 THEORY FOR BUCKLING OF ELASTICALLY SUPPORTED RING
29
B = dlmenflionless coefficient dependent, for the assumed elastic
ring, on the cylinder geometry and Polsson's ratio as shown
in Pig. 3.2 3^
Replacing Ea by 0.75 M, where M ■ constrained modulus of the soll (Luscher
and Hoeg, 1961*a, p. 122), leads to the following alternate form of Eq. (3.3):
K = 0.75 B M (3.^)
The constrained modulus M of medium to de'jse Ottawa sand was determined 0.8 In one-dlmenslonal compression tests as M = 1000 p . Substituting this
expression Into Eq. (3.M and then Eq. (3.1)» setting p = p* and solving
for p* resulted In the buckling equation
EI B \5/6
P* = 780 [ . ) (3.5) (- ■?
IMs expression correlated well with all buckling data Involving only
Insignificant arching (see Figure 7.35 of Luscher and Hoeg, 196^a).
The application of Eq. (3.5) to the failure conditions of smooth- walled tubes of dlf »rent materials assumed synoetrlcally surrounded by
the same soil with Infinite thickness leads to the curves of Fig. 3.3.
The figure demonßtiates that the failure stress In the elastic range Is
controlled by buckling curves similar In nature to column buckling curves.
At low slendemess ratios, these elastic buckling curves should be replaced
by curves for Inelastic buckling, which will be discussed In the next
section. However, already the elastic buckling curves simply truncated
by the horizontal yield stress line indicate, for various materials, the
approximate magnitude of the critical diameter-to-thickness ratios at
which the failure mode changes from cempressive yield to buckling.
The information on the critical diameter-to-thickness ratios for
3' it should be noted that the definition of B is different from that used in Luscher and Hoeg (196^).
30
1*
oo 0.80
(ro-ri)/rj
1.5 0.67 0.25
0.60
*•*
0.40
0.20
^ v = 0.3
v = 0^
s.
\ ^
0
B =-^_
0.2 0.4 n /r0
0.6 0.8 1.0
Es
[i- (n/rj2] (l + v) [l + (ri/r0)
2 (l-2v)]
FIG. 3.2 COEFFICIENT OF ELASTIC SOIL REACTION FOR ELASTIC RING
31
- . *
50,000
20P00
(A
10,000 -I-
t? (0 U) 01 k_ ♦- (D
0» _> CO (A
5,000
S 2,000 a E o o
o Ü
z ipoo
500
300 100 200 300 400
Radius to Thickness Ratio r/t 500 600
FIG. 3.3 BUCKLING STRENGTHS OF SOIL-SURROUNDED TUBES
Upper Limit
32
different materials is collected in Table 3»l»not only for the relation-
ship of Eq. (3«5) but axso for two similar relationships with reduced
K and therefore reduced buckling resistance, to investigate the effect S
of the quality of soil support. The specific reductions in K are by s
factors of 5,3 and 16, those in p* by factors of k and 10. (The physical
significance of these reductions is discussed in Section 3'7.) A tube
designed with D/t has a critical failure pressure of Just p* . If in
a given case the applied pressure multiplied by the desired factor of
safety is greater than p* , the required tube thickness will be larger cr
than t , therefore D/t will be smaller than D/t . and design on the cr' cr'
basis of yield stresses is permissible. If the design pressure is smaller
than the critical buckling p- «ssure p* , on the other hand, design on
the basis of yield stresses (ring compression theory) is unsafe, since
buckling rather than yielding will be the controlling failure mo3e. More-
over, the quoted critical radius-to-thickness ratios are upper limits,
since inelastic buckling was neglected; this is discussed in Section 3»3»
It is concluded from Table 3*1 that buckling will be the critical mode of
failure in many situations involving smooth-walled, flexible tubes and
should not be neglected in the design.
It should be noted at this time that the value for B is based
on formulation of the resistance of an elastic thick ring to inside
pressures uniformly applied, i.e., in a zero mode. Strictly speaking,
the resistance to infllde pressures with an n -mode distribution should
be associated with buckling in the n mode. This approach has been used
by Forrestal and Herrmann (1964) to arrive at buckling pressures of a long
cylindrical shell surrounded by an Elastic medium. The resulting buckling o
resistances, for the case of E/E = 10 ar.d V =0.3, are higher than s
those calculated on the bets is of Eq. (3*1) by factors 2.5 (at D/t = 100,
n = 7) to 10 (at D/t = 1000, n = 70).
However, Eq. (3.5) and hence Eq. (3«4) have been verified by
experiment. Furthermore, a relationship for K derived by Meyerhof and s
Baikie (19^3) on the basis of highly slmpJ-ifled concepts, and also veri-
fied by experiments, is practically identical to that of Fig. 3.2 for
infinite soil thickness.
33
r
n£
I
n
f
?
a
i o
CD a
n
ct o
s1 n
PtT
The author's use In the theoretical derivation of a soil resis-
tance "based on a zero rather than an n mode of deformation must be
considered a deviation from rigorous theory. Hovever, the theory agrees
well with experimental evidence, and is on the conservative side of the
rigorous theory while at the same time yielding theoretical buckling
values much In excess of what had been considered possible heretofore.
These facts make the theory an extremely useful tool, even if its deriva-
tion may include empirical steps.
3.3 INELASTIC BUCKLING OF SOIL-SURROUKDEDf SMOOTfl CYUUDERS
The curves shown in Fig. 3»3 represent the elastic buckling curves,
which are similar in nature to Euler's curves for column buckling. Like
the Buler curves, they are only valid in the elastic region and raust be
replaced by Inelastic buckling curves for stresses approaching yield.
How are these inelastic buckling curves obtained?
Based on Timoshenko and Gere (1961), Meyerhof and Baikle (1963)
proposed use of the interaction formula
where u = critical inelastic buckling stress cr
R, = yield stress
rr* = p* R/t, elastic buckling stress as discussed in the
preceding section.
This equation allows for buckling in the inelastic region as well as acci-
dental eccentricities and lioperfectlons. Two examplet of the resulting
buckling curves, marked M + B, are shown In Pig. 3»^' Considering the
ability of the soil to prevent buckling due to initial eccentricities, and
the different character of stress-strain curves for different materiala
(e.g., steel and concrete), this formula appears too crude for general
application, and is probably overconservative for many situations.
An alternate approach, coomonly applied to column buckling, is
35
.
1200
Diameter - to - thickness Ratio D/t
FIG. 3.4 INELASTIC BUCKLING CURVES
36
based on use of the tangent modulus at a given stress level in Ueu ot*
the modulus of elasticity In the buckling equation. The tangent modulus
E. can be determined graphically from the stress-strain curves. Tlmoshenko
and Gere (19^1) recommend Instead use of the equation
Et - %
l - C fV(ry
where c Is a coefficient depending upon the material. Nondlmenslonal
stress-strain curves for several values of c are shown In Fig. 3-19 of
the reference. Use of this equation with the reconanended values of c = O.96
for steel and 0 for concrete resulted In the curves marked T + G In Fig.
3.4, Stuesl (1955) states that graphical determination of the tangent
modulus Is Inaccurate, and recommends Instead an Iterative numerical pro-
cedure. His recursion formula applied to a typical stress-strain curve
"o- aluminum 5052-0 yielded the curve marked ST In Pig. 3.^. T^e stress-
: ^'ain curve would roughly correspond to a c of O.85.
It is interesting to note at this point that the T + 0 curve for
concrete with c = 0 coincides with Meyerhof's interaction curve for the
same material. Thus Meyerhof's method corresponds in effect to the assump-
tion of c = 0 for all materials and does not consider the different character
of the stress-strain curves of different materials in the calculation of the
buckling curve. It is felt that the methods which consider these differ-
ences by use of the tangent modulus in the buckling equation are superior.
Finally, Inelastic tube buckling curves were calculated on the
basis of column buckling curves. For Aluminum 5052-0, the ALCOA Hand-
book's straight-line inelastic column formula was transferred to Fig.
3.4 by plotting equal reductions from the elastic buckling stresses at
equal jlendemess ratios. (For example, if for a column with a slendemess
ratio of 75 the elastic buckling formula yielded 18,200 psi, the plastic
one 7,800 psi, and for the tube the elastic buckling stress was 18,200 vsl
at R/t = 200, then § = 78OO pil atthatR/t.) The resulting curve,
marked AL in Pig. 3.h, is practically straight and represents an only
37
t
slightly conservative approximation to the curve based on the tangent
modulus. By the same method, Tetmajer's straight line for steel (Tlmoshenko
and Gere, 19^1), modified to account for a lover yield stress, gave an
Inelastic huckllng curve (marked TE in Fig. 3«^) very close to the curve calculated "by the c-formula.
As protection against uncertainties In the soil support, a reason-
ably concervatlve choice of the soil modulus K should be made (see Section
3.7). Then no additional conservative assumptions are required apart from
any desired uniform factor of safety Independent of D/t. Consequently it
is recannended that an inelastic buckling curve based on the tangent modulus
of the structural material be used.
An estimate can be made of the maximum amounts by which the in-
elastic buckling curves lie bei' *. composite line formed by the elastic
buckling curve and the horizontal yield stress line. This reduction is
a function mainly of the stress-strain curve of the material. It is smallest
for steel (15^). larger for aluminum (23%), largest for concrete (50^),
among the materials Investigated here. An empirically fitted relationship
for the maximum reduction, which occurs at the intersection of the elastic
buckling curve with the yield stress line, is
K, r- =1-0.5 Vl^T . (3.7)
Knowing this point and the elastic buckling curve and yield stress line,
which are both approached asymptotically, one can plot approximate inelastic
buckling curves.
3.4 BUCKIJNG OF FULLY BURIED CYUNDERS
So far only buckling of a circular-symnetric tube-soil configuration
has been considered. However, the situation of a buried cylinder (Fig. 1.1c),
loaded either by a deep fill or by a surcharge applied at the soil surface,
is not circular synmetric. The burled cylinder is deformed, with a horizontal
diameter appreciably larger than the vertical diameter, and with soil
38
*r
pressures not equal all around. These conditions may affect the buckling
resistance. Further, arching and pressure redistribution have to be con-
sidered in the desired expression relating the critical surface pressure
causing buckling to the pertinent parameters of the system. This section
investigates the situation of "full" burial, i.e., a depth of cover of at
least ID (Hoeg, 1965; Donne Han, 1964) where the effects of the soil
boundary became negligible. The subsequent section investigates the effect
of the depth of cover.
A first correction to be applied to the buckling equations, Eqs.
(3.1) or (3»5)» is one to account for pressure redistribution and arching.
As concluded in Section 2.2, the pressure acting on the tube amounts to
roughly two-thirds of the pressure applied on the surface of dense soil,
and to the full value of the surface pressure for loose soil. Assuming
now that the pressure causing buckling is unchanged from the circular-
synmetric situation, the resulting equation for P* > the surface pressure
causing elastic buckling, is
K El
p*o= 2?i~^r~ ' (3-8)
where, as before, [3 = 1,5 for dense soil and 1.0 for loose soil. The same
modification applies to Eq. (3.5) for dense Ottawa sand:
E! B • 5/6 / El B v 0.9)
The second effect on buckling resistance to be investigated here
is that of the deformation of the cylinder and the resulting non-uniformity of
pressures acting on it. For purposes of this investigation, it is assumed
that the buckling equations apply locally rather than for the tube as a
whole. I.e., for local radii and local contact pressures. Further, it
is assumed that the tube deforms according to the concepts outlined In
Section 2.3 .id that the deformations obey Eq. (2.3). Finally, it is
39
*
assumed that the hoop compresslve resultant N equals R o/ß , the tube
radius times the applied surface pressure modified by the arching factor
[3 , and that the local contact pressure can be computed as N divided by
the local radius.
Under these assumptions, the crown and invert of the cylinder
become the critical buckling locations. Substitution of expressions for
the local radius and contact pressure at these locations into Eq. (3«9)
and neglect of higher order terms leads to an expression for p* , the sur-
face pressure to buckle a deformed tube:
p*e = p*o (1 - U.5 Ej, (3.10)
Using Eq. (2.3) and neglecting the effect of the tube rigidity on defor-
mation, this equation is transformed to
/ 3-5 P*e \ p» = p* 1 S_) (3.11) e 0 l /3 E* '
Equations (3.10) and (3.11) express the dependency of the critical buckling
pressure upon the tube deformation.
Two realistic expressions for E* can now be used to explore numeri-
cally the effect of £,. upon p* . First, it is assumed that E* = 3000 psi,
which is a reasonable value for a good backfill according to Section 2.3«
Then,
p» = p* (1 - 0.0008 p* ).
A second expression is the one backcalculated for laboratory-placed dense
sand, E* = 2000 pf. Then E* ^_ = 1000 (p» )5, and • average e
i. P*e = P*0 (1 - 0.00233 (P*e)2).
Typical values of reductions in buckling pressure obtained by the two
kO
equations are the following:
E* = 3000 psl E* = 2000 1
P*o psl
P*e psl
reduction *
P*e psl
reduction
10 9.9 1 9.9 i
20 19 6 2 19.8 i
100 93 7 93 2
500 357 29 U75 5
It Is seen that the constant E*-value gives Increasingly large
reductions with Increasing pressure level. In the case of the variable
E*, the reduction Increases more slowly with the pressure level, since
now the deformation Is proportional to only a fractional power of p* . The
result Is more realistic for dense, I.e., locking aand.
Ttie conclusion from the ahove Is that deformation of the buried
tube decreases the buckling resistance In dense soil inappreciably.
Thus the buckling expressions, Eqs. (3.8) and (3.9)» can in first approxi-
mation be directly applied to this situation. For loose backfill, however,
for which the E* may be only a fraction of the 3000 psl assumed above,
the effect of deformations can became appreciable. For instance, for the
recommended maximum design deformation of 5^ for burled cylinder« (Spangler,
i960), which will ordinarily be reached only in loose soil, the reduction
is 22^.
Experimental verification of these concepts is scarce, since the
effect Is always superimposed on, and is secondary to, the general increase
in buckling resistance due to the surrounding soil. Further, the uncertain-
ties In the amount of arching present are generally at least as large as
the effect Investigated here. For instance, the data by Luscher and Höeg
(I96^a), which were already quoted in support of arching concepts, allow
conclusions supporting the concepts advanced here only if the arching con-
cepts are assumed correct. Under this assumption the data indicate that
for these tubes the reduction in buckling pressure due to deformation was
negligible, as predicted by the theory for tubes burled In dense sand.
hi
> t
The conclusion, therefore, 18 that the main effects of burial to
be considered are pressure redistribution and arching. The effect of
deformations on buckling pressure can In general be neglected In view of
the much larger uncertainties associated with the choice of the modulus
of soil rigidity K . Equations (3.8) and (3.9) are the modified buckling s equations to be used for burled cylinders.
3.5 BUCKUNG OF SHALLOW-BURIED CYUNDERS
The preceding section explored the effect on buckling of burial
at a depth of at least ID, where the effects of the soil boundary become
negligible. In this section the effect of a nearby boundary on the buckling
resistance of a burled tube is to be investigated. The loading is applied
as pressure on that boundary.
Three effects might tend to reduce the buckling resistance in
this situation. The major one is a reduction in the coefficient of soil
reaction K , which is responsible for the tremendous increase in buckling s strength of a soil-surrounded tube over the unsupported tube. The second
effect is an increase in deformation of the tube due to a loss in E* close
to the surface. It is felt that this effect is much less important than
the first one and can therefore be neglected. The same is true of the
third effect, a possible reduction in arching and pressure redistribution
with decreasing depth of cover. This effect is thought to be relatively
small on the basis of the data collected in Section 2.2 and Hoeg's (1965)
and Donnellan's (1964) results.
Luscher and Höeg (196^) have studied the variation of the coeffi-
cient of soil reaction K with soil thickness for the circular-symmetric
tube-soil configuration. They found that the coefficient B in Eq. (3*3)
or {3»h), determined theoretically and plotted versus the ratio of the
outside to the inside radius of the soil cylinder In Fig. 3«2, correctly
described the variation for that case.
A conservative extension of this finding to the burled-cylinder
situation is to use the cover over the crown as thickness of the equivalent
h2
«•-....
soil cylinder. In other words, Pig. 3.2, with the abscissa changed to
R/Cd-R) (i.e., tube radius divided by depth of tube center), is directly
applied to the burled-tube situation. This formulation is conservative
because it vises as equivalent outside radius the shortest distance to the
surface, which occurs at one point only. Ideally the average distance to
the surface over a certain central angle, which might be related to the
expected failure mode, should be chosen as outside radius. The need for
such a modification is seen most clearly in tests where a tube with exposed
crown still shows much higher buckling strength than an unsupported tube.
For the sake of simplicity ar-d conservatism, however, such a modification
is not used here.
Two sets of failure data of tubes buried at shallow depth are avail-
able to check this theory. The first of these «re the data from tests on
two-ply aluminum foil tubes of 1.6-in. diameter buried in dense Ottawa sand,
reported in Fig. 8.3 of Luscher and Hoeg (l96Ua) and reproduced in Fig. 3.5
of this report. Also shown on the figure are the buckling pressures pre-
dicted by Eq. (3.9)> with B from Fig. 3.2. The agreement between theory
and experiment, particularly in legard to the variation of buckling pres-
sure with depth, is seen to be satisfactory.
The second set of applicable data are found in Table 1 of Donnellan
{196k) for buckling of shallow-burled aluminum cylinders of U-ln. diameter
and D/t varying between l60 and hOO. These data, together with theoretical
predictions of buckling pressure, are given in Table 3-2. For the theoret-
ical prediction, E and therefore K were assumed to be 25^ higher than s s
in the M.I.T. tests, since the E*-values backcalculated from tube defor-
mation data (Table 2,2) were so related. Further, the buckling pressures
in the Inelastic range were determined from the available curve for Alumi-
num 5052-0 with a yield stress of 10,000 psi (Fig. 3.10* although the
material of these tubes was Aluminum 2024-0 with a yield stress of only
8000 psi. Finally, the factor used to account for arching and pressure
redistribution was not 1.5 as recommended in the preceding section but
2.0, in better ekgreement with factors actually observed in instrumented
tests described in the same reference.
"•a
•tv
•fV
Failure Pressure - psi
P
n c -
M r >
C
G 33
33
n T: r z o
s
E/3
r r
O
"3 H
i? «-»■ 3*
0
33 c
a'
to
re 73
-1 » a c' 7)
10 33
33
:r 3-
TB "CD
ii II
^
! t; ro ro UJ ö ON rt U1 OJ Q
i 0 o UJ 1 **
! v a H »
1 0 £ ro 00 fc ii
1 ! ^ ; H H H H
^ 1 v^ S" Ul UO •p- I o UJ F ro CO H
1 1 O i •d o 1 CO I—) , ! v V
1 H H H H V^l KS\ U) VJ1
^ -\ •d
1 O i p
O ->! ro
•
CO 1 <* j
v V V V V
V
3 1 H h-1 H H» H v CO
VJ1 ^ \J\ VJ1 VI 0) | O o O O O ro i
1
p< 1 \J\ ro H O o a* • 1 ^
t
CD • ■P-
• • n o s
(9 "•
H- 1 O ! ro H H 0* 1 ^ OS
CD § V/1 M !
1 i " '1
to H H ! t i
i U) H § CO ^ ^1 ■P- j ^ !
'V i i m
M H H H | m g ^ 8. d "ro 1
i
\
1' ■-
ro H M H i
^ H 5 ^ 8:
i
ro
The correlation observed In Table 3»2 iß quite good, especially
for the shallover depths of cover. For deeper depths, the theory under-
estimates the buckling strength, probably because of an underestimate of
arching In this exceptionally dense sand. For zero depth, the buckling
strength Is much higher than for an unsupported tube for reasons g-'ven
earlier. This case docs not have much practical significance and Is not
covered by the theory.
It is concluded that the effect of depth of cover can be considered
by varying the coefficient B In Eq. (3.9) according to Fig. 3.2, treat-
ing the depth like the thickness of a surrounding soil cylinder.
3.6 BUCKUNG RESISTANCE OF CORRUGATED PIPES
Corrugation of pipes has been used for a long time as a means
of increasing thair flexural rigidity for handling, backfilling and better
performance purposes. Since buckling has been found to be the controll-
ing mode of failure for many smooth-walled cylinder InstaLTatlons, it
appears of Interest to investigate the buckling resistance of corrugated
pipes.
The point of primary interest on the buckling curve is again the
point of Intersection vith the yield-stress line. The "critical" clameter
corresponding to this point has been calculated by Eq. (3.5) for two
steel thicknesses, 10-gage (O.I3U5 in.) and l6-gage (O.0598 in.), and for M different standard corrugations as well as for a smooth plate. / Addition-
ally the critical diameter for a buckling curve that Is lower by a factor
k (i.e., K reduced about five-fold) has been determined to investigate s the effect of the quality of soil support. The results of these calcula-
tlonr. are presented in Table 3.3»
_
For simplicity, arching and pressure redistribution have been neglected in this calculation, as Indicated by the use of Eq. (3.5) Instead of (3*9). The conclusions will therefore be conservative for dense soil and about right for loose soil.
k6
TABLE 3.3 BUCKUNG RESISTANCE OF CORRUGATED PIPES
D (in.) at Intersection er of elastic buckling curve
with yield stress line "by Eq. {3*1^)
Corrugation
Upper
p* = 78O
limit
/ EIBN5/6
Practical
P* = 195 |
mininrum
'al)5/6 [ R5) ^)
10-gage
1500
700
16-gage
(1630) not used
760
10-gage 16-gage
Multi-plate 2 x 6 in.
(650) 600 not used
Co-rugation 1 x 3 in. 280 300
Corrugation i x 2-2/3 in. 3U0 370 13^ 150
Smooth pipe ' 80 38 32 111
1) These values are for 8- and 14-gage smooth pipes, which have about the same cross-sectional area as 10- and l6- gage corrugated pipes, respectively.
^7
r
It is seen from the table that corrugation increases the critical
diameter many-fold over that of a smooth tube of similar cross-sectional
area. The increases are sufficient so that probably all critical diameters
are beyond practically considered use, even in the case of the reduced
soil support. It is therefore concluded that the ring compression theory,
with some allovance for inelastic buckling at larger diameters, is valid
for the large majority of corrugated pipes.
3.7 EF'PECT OF SOIL STIFFNESS AND PORE PRESSURE ON TUBE BUCKLING RESISTANCE
3.7.1 Effect of Soil Stiffness
So far in this report Eqs. (3.5) and (3.9) have been used when-
ever numerical values of calculated buckling pressures were needed. The
dependency of K upon the confining pressure Implied in these equations s
was experimentally determined in tube buckling tests. The dependency was
also successfully related by theory to soil modulus values determined in
one-dimensional compression tests. This correlation between K and M s allows investigation of the dependency of the buckling resistaftbe upon
the soil stiffness by use in the theory of constrained moduli for the
soil of interest. Substituting Eq. (3.^), Ka = 0.75 B M, into Eq. (3.ß) s leads to the controlling expression
if EI B r— p*o = 1.73 p I 3— IM. (3.12)
I R
The modulus M is usually determined in one-dimensional laboratory
compression tests. An alternate way, for buried pipes, is to use M-values
backcalculated from measured deformations of tubes (see Section 2.3, in
particular Eq. (2.5)). This method may appear to be less direct, since
M is related to E* by quite crude mathematics. However, for use in cal-
culating K , which is a property of the soil next to the tube Just like S
E* and hence closely related to E*, this method of determining M appears
appropriate. The direct relationship between K and E*, calculated for s
infinite depth from Eqs. (2.5) and (3.4) and Fig. 3.2, is
Kg = 2.5 E* (3.13)
k8
Any error in this relationship should be systematic and subject to
elimination by direct tests involving both properties.
Thus the task of choosing reasonable values of K is reduced to s
the task of establishing reasonable values of M from one-dimensional com-
pression tests, tube deformation data or tube buckling data. This is done
with the help of Pig. 3*6, vhich is a summary plot of M-values from many
sources. Whitman (1964) was particularly helpful in pointing out sources
of compression test data. The M-values derived from tube deformation data
were taken from Section 2.3. The following is concluded from the figure:
1. The M-curve backcalculated from tube buckling tests in
dense sand (Luscher and Hoeg, 196Ua) represents a reason-
able upper limit of the modulus. Higher values can be
achieved particularly in the laboratory, but are probably
unrealistic for practical situations. Thus Eqs. (3*5)
and (3»9); which are based on this relationship, give
reasonable upper Unite of buckling resistance.
2* An M-curve 16 times lower than the upper limit gives a
reasonable lower limit which should be conservative in
most cases. The 16-fold reduction in M results in an
approximately 10-fold reduction in K (with the 5/6-power s
law used for p*). Lower limits may be expected for the
one case of tubes driven or tunnelled into soft clay.
3. The "reasonable lower limit" of K should never be s
reached for carefully installed pipes. A value of K
lowered by a factor of approximately five, giving a
four-fold reduction in elastic buckling stress compared
to the reasonable upper limit, appears to be a reasonable
practical minimum for high-grade Installations. This
reduction has been used in the preceding sections as an
example of reduced soil support.
h9
T
0^
Constrained Modulus M - 10 psi
en o
TTTT
-r o •
r
H i—i 2: O < > r n M
n c
>
M Ö
C
r G CD
o o o
The above information is summarized by the equation
P* =AK EIB ^5/6
,3 I (3.1M
where A = 780 for "reasonable upper limit,"
A = 195 for "practical minimum,"
A = 78 for "reasonable lower limit."
Application of this equation to the "critical" buckling condition at which
the failure mode changes from compressive yield to buckling is shown in
Table 3«1« The critical diameter-to-thickness ratio is lowered by a
factor 2.5 for the "practical minimum" compared to the upper limit of K . s
For the "reasonable lower limit" this reduction would be as high as 4.64-
fold. Probably Just the lowering of the ratio to the practical minimum,
and certainly the lowering to the reasonable lower limit will bring most
metallic tubes within the range where buckling rather than compressive
yield is the controlling failure mode. This fact Justifies and motivates
adoption of corrugated cylinders for more efficient use of the structural
material.
In practical cases, K should be chosen on the basis of one- s
dimensional compression tests and substitution into Eq. (3«^)» Alter-
natively, highly approximate values can be directly selected from the
information presented in this report.
3.7.2 Effect of Pore Pressures
Another effect influencing the buckling resistance is the presence
of pore pressure. This effect was studied by Luscher and Hoeg (1964a) in
tests In which part of the pressure acting on the tube was effective pres-
sure p transmitted through the soil, and part was pore pressure u. The study
establlsned that only the effective stresses contributed to buckling re-
sistance, while the neutral stresses contributed only to applied pressure.
Thus the total pressure causing buckling can be formulated as
p + u = 2 K El s
(3.15)
51
.- ■ ■ t
where now K Is a function of p only.
For the calculations exploring the effect of a given pore pressure,
It is advantageous to work with a K and consequently em M which are pro- - ^ n ft
portlonal to p rather them to the (p) * used to derive Eqs. (3.5) emd (3.9).
Formulating M = m p , where m Is the value of the M-function for p = 1 psl,
and substituting Into Eq. (3.15) results In
EI B p i u = 2 y—j- fa p .
Calling p* the solution which would be obtained for zero pore pressure,
^ EI B m p* =
R3
the equation for the effect of u becomes
Jpl 1 1-^ - u- ] ' (3.16) P =
Application of this equation leads to the Interaction curves shown 5^
In Fig. 3»7. It Is seen that there eure two possible fetllure conditions
for each value of pore pressure, the lower one leading to a total buckling
pressure p + u of less than 50 percent of the "dry buckling pressure" p*,
the upper one to more them 50 percent. Which solution Is controlling In
a particular case will depend on the condition of loading. The lower part
of the curve Is valid In situations where the pore pressure Is larger than
half the total pressure, as Is for Instance the case In shallow tunnels
under harbors and rivers. In protective construction, where pore pressures
will generally be small or nonexistent, the upper part of the curve will
apply. Consequently the reduction In total buckling pressure will remain
''' Strictly speeiking, the curves for (p + u)/p* and u/p* should not go to zero for p/p* = 0, but to the free-air (hydrostatic) buckling pressure. This limitation is the same as discussed in Footnote 2) in Section 3.2.
52
(a
C ed
I* 3 O-
0.5
1.0
P "I P
FIG. 3.7 EFFECT OF PORE PRESSURE ON BUCKLING RESISTANCE
53
less than 10 percent as long as the pore pressure Is less than 10 percent
of p*. The reduction reaches alnost 30 percent for a pore pressure of 20
percent of p*, and Is 50 percent for the maximum possible pore pressure
of 25 percent of p*.
It is concluded that small pore pressures have relatively little
effect upon buckling pressure as long as the effective stress Is higher
than the pore pressure, as Is generally the case In protective construction.
However, If the reverse Is true, the tube buckling pressure can be reduced
to a small fraction of the "dry buckling pressure" p*.
5»*
CHAPTER h
SUWARY AND CONCLUSIONS
h.l SUMMARY
This report treats the "elastic" "behavior and the failure con-
dition of underground flexible cylinders. Specific aspects treated in
Chapter 2 include the loads reaching the burled cylinders, the deforma-
tions undergone by the cylinder as the load is applied, and the possible
failure modes. The most important conclusions were.'
1. Arching was not found to be a very important factor for
flexible cylindrical structures with horizontal axis
buried In soil at depths up to several cylinder diameters.
Active arching may reduce the load applied en cylinders
buried in stiff soil by up to 30 or possibly 50 percent.
Inversely, passive arching may subject cylinders burled
in compressible soil to loads somewhat higher than applied
on the soil surface. These relationships are expressed
in the highly approximate equation
P = Po/p ; (2.1)
where ß = 1.5 for stiff soil, 1.0 for compressible soil.
2. A modification of Spangler's classical equation for the
calculation of pipe deformations is proposed, to take account
of arching and to employ more realistic concepts of lateral
pressures acting on the pipe:
r A ADh 0-5/ß ' p0 A£.*A~-2i-* iT-5 (2.3)
85 EI/IT + 0.65 E»
Application of this equation in increments allows introduction
of an E* that varies with pressure.
55
."
3* By use of Eq. (2.3) values of the modified modulus of passive
soil resistance B* were backcalculated from available cylinder
deformation data, and were correlated to constrained moduli M
of the soils "by the equation
B» = 0.45 M. (2.5)
h, Corapresslve yield (or inelastic buckling at only slightly
reduced pressure) is the desirable failure mode. To achieve
it, the various modes of premature failure have to be avoided.
Buckling of underground flexible cylinders is treated extensively
in Chapter 3, in continuation and termination of a study initiated several
years ago. From the simple circular-symmetric situation analyzed earlier,
the investigation was extended to underground geometry, to inelastic
buckling, to corrugated pipes, and to the effects of soil stiffness and
the presence of water. Ihe main results were:
5. Theoretical equations for elastic buckling of the synnnetrir
soil-tube configuration were established not only in terms of
em unknown coefficient of soil reaction K - s
p» = 2 KEI -V- (3.1)
dency of K upon the applied pressure, valid for dense sand: S
- but also in terms of an experimentally determined depen-
iBBVa
EIB \5/6
P* - 780 [ y-l (3.5) 7 Furthermore, K was correlated to the one-dimensional soil
modulus M:
K = 0.75 B M , (3.M 0
where B is a coefficient obtained from Fig. 3*2.
56
6, The bucMlng theory was extended to th« inelastic range by
vuse of the tangent modulus of the structural material in
the buckling equations. A procedure was proposed for simple
determination of an approximate inelastic buckling curve,
fitted between the horizontal yield stress line on one side
and the elastic buckling curve on the other side.
?. Ttie effect of burial was considered by including in the
buckling equations a factor ß to account for arching, and by
using the theoretical dependency (expressed in Pig. 3.2) of K upon the soil cover over the cylinder: s
K El P*0 = 2P1I % (3.8) o ry R3
5/6 p*o = 78o|3 (-JL*-) (3.9)
R
The effect of tube deformations upon buckling was found to
be relatively minor.
6. While for many smooth-walled, flexible metallic tubes elastic
buckling is the controlling mode of failure, corrugation in-
creases the buckling resistance to a point where inelastic
buckling or compressIve yield usually are the controlling
fed.lure modes.
9. The effect of soil stiffness was considered by establishing
a reasonable range of possible values of K_. Use of the s correlation between K and M.as well as the correlation be-
s ' tween E* and M, resulted in the buckling expression
5/6 P*o = ^(-^T-) ^-^
where A = 780 for "reasonable upper limit,"
A = 195 for "practical minimum,"
A = 78 for "reasonable lower limit."
10. The presence of pore pressure reduces the buckling resistance
only inappreciably, as long as the pore pressure is consider-
ably smaller than the effective stress.
57
^.2 COLLUSIONS REGARDIKG THE IMPORTANCE OF TOBE BÜCKLING
When good backfill procedures are used, the expected critical
D/t-ratlos at vhlch the failure mode changes from compresslve yield to
huckllng are as given in Table 3«1 under "practical minlatum." The quoted
values place many smooth metallic tubes into the region where buckling
failure is controlling, and where consequently the buckling theory presented
in this report should be applied. Further, even if D/t is only approach-
ing the critical value, inelastic buckling and the effect of tube deforma-
tions reduce the critical pressure below yield. This, and the uncertainty
associated with arching, forbid i&e of the full yield stress in design even
below the critical D/t. For ü/t above the critical value, the buckling
pressure approaches asymptotically the calculated elastic buckling stress.
Corrugation of the cylinder material is an extremely effective means
of increasing the buckling resistance, so that elastic buckling should
only in extreme cases be controlling. The knowledge of how close one is
to the critical condition In a given situation is helpful in evaluating
the required factor of safety with respect to yield.
The correlation of K with M or E*, and the methods of determining s
K from one "dimensional compression tests or from the information presented s
herein, can presently be considered only rough approximations. Hopefully,
more data will soon become available to improve the method of selecting
K for a given situation, s
58
REFERENCES
Allgood, J. R. (1965) The Behavior of Shallov-Burled Cylinders - A Synthesis and Extension of Contemporary Knowledge, Technical Report R 3^ of the U. S. Naval Civil Engineering Laboratory. Also in Proceedings of the Symposium on Soil-Structure Inter- action, Tucson, Arizona, Sept. 196^.
ASCE (196^) "Development and Use of the Modpares Device," Research Council on Pipeline Crossings of Railroads and Highways, Proc. ASCE, Vol. 90, No. PL1, Jan.
Barnard, R. E. (1957) "Design and Deflection Control of Burled Steel Pipe Supporting Earth Loads and Live Loads," Proceedings of the ASTM, Vol. 57, PP. 1233-1256.
Bulson, P. S. (1962) Deflection and Collapse of Burled Tubes, Report RBS 7/l of the Military Engineering Experimental Establish- ment (MEXE), England.
Demmin.J. (196^) Field Verification of Ring Compression Conduit Design, Report of Armco-Thyssen, Dinslaken^ Germany. ~
de Souza, J. M. (1959) Compressibility of Quartz Sand at High Pressure, S. M. Thesis, M.I.T. Department of Civil Engineering.
Donnellan, B. A. (I96U) The Response of Burled Cylinders to Quasi- Static Overpressures, Report No» AFWL TDR-64-13. Also in Proceedings of the Symposium on Soil-Structure Interaction, Tucson, Arizona, Sept. 1964.
Finn, W. D. L. (1963) "Boundary Value Problems of Soil Mechanics," Proceedings ASCE, Vol. 89, No. SM5, Sept.
Forrestal, M. J., and G. Herrmann (1964) "Buckling of a Long Cylindrical Shell Surrounded by an Elastic Medium," ASCE Structural Engineer- ing Conference and Annual Meeting, October, Conference Preprint 108.
59
Hasslb, (1951) ConaoHdatlon Chai^icterlstlcs of Granular SolLs, Columbia University, New.York. "~^~'
Hendron, A. J. (1963) The Behavior of Sand in One-Dimensional Compression, Report No. AFWL TDR-63-3OÖ9. Also Ph.D. Thesis, University of Illinois Dept. of Civil Engineering.
Höeg, K. (1965) Pressure Distribution on Underground Structural Cylinders, Report No. AFWL TR b^-90.
Lambe, T. W. (i960) Buried Flexible Pipes, Report for the Algonquin Gas Transmission Company, Boston, Mass.
Lambe, T. W. (1963) "An Earth Dam for the Storage of Fuel Oil," 2nd Pan- American Conference on Soil Mechanics and Foundation Engineering, Brazil.
Link, H. (1963) "Beitreg zum Knickproblem des elastisch gebetteten Kreisbogenträgers," Stahlbau, Vol. 32, No. 7, July, pp. 199-203.
Luscher, U. (1963) Study of the Collapse of Small Soil-Surrounded Tubes, Report No. AFSV^ TDR-S^^
Luscher, U., and K. Hocg (196^) The Interaction Between a Structural Tube and the Surrounding Soil, Technical Documentary Report No. AFWL TDR-63-3IO9.
Luscher, U., and K. Hoeg (1964b) "The Beneficial Action of the Surrounding Soil on the Load-Carrying Capacity of Buried Tubes," Proceedings of the Symposium on Soil-Structure Interaction, Tucson, Arizona, September, pp. 393-h02t Also M.I.T. Civil Engineering Department Professional Paper No. ?6h-2h, May I96U.
Luscher, U., (1965) Static and Dynamic Tests on U-Inch-Diameter Cylinders, Report No. AFWL TDR 64-92, in preparation. ~~
Marino, R. L., Jr., (19^3) A Study of Static and Dynamic Resistance and Behavior of Structural Elements, Report No. AFWL TDR-63-306O.
Meyerhof, G. G., and L. D. Baikie (1963) "Strength of Steel Culvert Sheets Bearing Against Compacted Sand Backfill," Highway Research Board Record No. 30, pp. 1-14.
Newmark, N. M. and J. D. Haltlwanger (1962) Air Force Design Manual, Report No. AFSWC TDR-62-138 Confidential). """
Robinson, R. R., (1962) The Investigation of Silo and Tunnel Linings, Report No. AFSW: TDR-62-1.
Spangler, M. G., (19^0) Soil Mechanics, Int. Textbook Co.
Stüssl, F., (1955) Tragwerke aus Aluminium, Springer-Verlag.
6C
Tlmoshenko, S., and J. Gere (1961) "Rieory of Elastic Stability, McGrav-HiU.
watklns, R. K., (1959) "Influence of Soil Characteristics on the Deformation of Qnbedded Flexible Pipe Culverts," Highway Research Board Bulletin 223, PP. 1^-24,
White, H. L., (1961) "Largest Metal Culvert Designed by Ring Compres- sion Theory," Civil Engineering, Vol. 31, January, pp. 52-55.
White, H. L., and J. P. Layer (i960) "The Corrugated Metal Conduit as a Compression Ring," Highway Research Board Proc.
Whitman, R. V., (l^) "Mechanical Properties of Soil," Nuclear Geoploslcs, Part Two - Mechanical Properties of Earth Materials, DASA-12Ö5 (II).
Whitman, R. V., and U. Luscher (1962) "Basic Experiment Into Soil- Structure Interaction," Proc. ASCE, Vol. 88, No. SM6, Dec.
Whitman, R. V., U. Luscher, and A. R. Philippe (1961) Preliminary Design Study for a Dynamic Soil Testing Laboratory, Final Report for Contract AF 29(601)-19U7. Report No. AFSW: TR-6I-58 contains Appendices K, L, M and N to this report.
Yoshimi, Y., and J. 0 Osterberg (1963) "Compression of Partially Saturated Cohesive Soils," Proc. ASCE, Vol. 89, No. SM^, July.
61
BLANK PAGE
UNCLASSIFIED Secunty Classification
DOCUMENT CONTROL DATA • R&D (Security clatmilicalion ol title body ol abatracl and indexing annolalion mual be entered whan Ihr overall report i« c laamilied)
1 ORirilNATINGACT|v/|TY (Corpmrale author)
Massachusetts Institute of Technology, Department of Civil Engineering
Cambridge 39, Massachusetts
2a RCOORT SECURl TY CLASSIFICATION
UNCLASSIFIED 2 t> GROUP
3 REPORT TITLE
BEHAVIOR OF FLEXILLE UNDERGROUND CYLINDERS
4 DESCRIPTIVE NOTES (Type ol report and inclusive datea)
1 March 196^-2 April 1965 5 AUTKORfS; rL«»f name, f/r«( name, Initial)
Luscher, Ulrich
6 REPORT DATE
September 1965 7a TOTAL NO OF PACES 76 NO OF REFS
lh -L
33
aa CONTRACT OR GRANT NO ^p- 29 ( 601 )-6368
5710
9a ORIGINATOR'S REPORT NUMBEBfS)
6, PROJEC T NO AFWL-TR-65-99
c DASA Subtask No. 13.157
d
9b OTHER REPORT NOCS; (Any other number» that may be aeelgned thla report)
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DDC release to OTS is authorized.
11 SUPPL EMENTARY NOTES 12 SPONSORING MILITARY ACTIVITY
AIWL (WLDC) Kirtland ATB, NM 87117
is ABSTRACT fa investi gat ion was made of the "elastic" behavior and failure condition of underground flexible cylinders with particular attention given to arching, defor- mation and buckling. The report presents no new data, rather draws heavily from ex- perimental and theoretical work done in the past several years in an attempt to ar- rive at a unified picture of the chosen aspects of behavior. Active arching was found to reduce the load acting on tubes buried at depths up to several diameters in stiff soil by an average of 30 percent. On the other hand, passive arching may sub- ject tubes buried in compressible soil to loads somewhat higher than applied on the surface. Spangler's deformation equation was modified to account for arching, later- al pressures, and variability of the soil modulus with pressure. Values of the modi- fied modulus of passive soil resistance, backcalculated by the new equation from tube defonnation data, were successfully related to the constrained modulus of the soil. A comprehensive theory of buckling of underground cylinders is presented. It starts with the previously derived theory for elastic buckling in the circular-sym- metric tube-soil configuration and extends it to cover (l) elastic buckling of an underground cylinder; (2) inelastic buckling; (3) the effects of soil stiffhess and presence of water; and (1*) buckling of corrugated cylinders. It proved possible to correlate the soil modulus Kg controlling buckling to the constrained modulus M of the soil. The theory agreed well with the few available data. More comparisons with laboratory and field data are required, in particular to verify the values of Ks and their relationship to values of M. Regardless of the exact value of K8, however, it was shown that for many practical situations of underground cylinders the controlliig modp of failure is buckling rather than corapressive yield.
DD FORM I JAN e4 1473 UNCLASSIFIED
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UN CLASS IFTF.n Security Classification
14 KEY WORDS
Behavior of flexible pipe Buried cylinder research Soil-structure interaction Horizontally buried cylinders
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